NAG Toolbox: nag_lapack_zggesx (f08xp)

Purpose

nag_lapack_zggesx (f08xp) computes the generalized eigenvalues, the generalized Schur form (S,T)(S,T) and, optionally, the left and/or right generalized Schur vectors for a pair of nn by nn complex nonsymmetric matrices (A,B)(A,B).

Estimates of condition numbers for selected generalized eigenvalue clusters and Schur vectors are also computed.

Description

The generalized Schur factorization for a pair of complex matrices (A,B)(A,B) is given by

A = QSZH, B = QTZH,

A=QSZH, B=QTZH,

where QQ and ZZ are unitary, TT and SS are upper triangular. The generalized eigenvalues, λλ, of (A,B)(A,B) are computed from the diagonals of TT and SS and satisfy

Az = λBz,

Az=λBz,

where zz is the corresponding generalized eigenvector. λλ is actually returned as the pair (α,β)(α,β) such that

λ = α / β

λ=α/β

since ββ, or even both αα and ββ can be zero. The columns of QQ and ZZ are the left and right generalized Schur vectors of (A,B)(A,B).

Optionally, nag_lapack_zggesx (f08xp) can order the generalized eigenvalues on the diagonals of (S,T)(S,T) so that selected eigenvalues are at the top left. The leading columns of QQ and ZZ then form an orthonormal basis for the corresponding eigenspaces, the deflating subspaces.

nag_lapack_zggesx (f08xp) computes TT to have real non-negative diagonal entries. The generalized Schur factorization, before reordering, is computed by the QZQZ algorithm.

The reciprocals of the condition estimates, the reciprocal values of the left and right projection norms, are returned in rconde(1)rconde1 and rconde(2)rconde2 respectively, for the selected generalized eigenvalues, together with reciprocal condition estimates for the corresponding left and right deflating subspaces, in rcondv(1)rcondv1 and rcondv(2)rcondv2. See Section 4.11 of Anderson et al. (1999) for further information.

alpha(j) / beta(j)alphaj/betaj, for j = 1,2, … ,nj=1,2,…,n, will be the generalized eigenvalues. alpha(j)alphaj and beta(j),j = 1,2, … ,nbetaj,j=1,2,…,n are the diagonals of the complex Schur form (S,T)(S,T). beta(j)betaj will be non-negative real.

Note: the quotients alpha(j) / beta(j)alphaj/betaj may easily overflow or underflow, and beta(j)betaj may even be zero. Thus, you should avoid naively computing the ratio α / βα/β. However, alpha will always be less than and usually comparable with ‖a‖‖a‖ in magnitude, and beta will always be less than and usually comparable with ‖b‖‖b‖.

It is possible that info refers to a parameter that is omitted from the MATLAB interface. This usually indicates that an error in one of the other input parameters has caused an incorrect value to be inferred.

After reordering, roundoff changed values of some complex eigenvalues so that leading eigenvalues in the generalized Schur form no longer satisfy selctg = trueselctg=true. This could also be caused by underflow due to scaling.